Image denoising method based on hyperparameter-free total variation equilibrium constraint

Abstract

The application discloses an image denoising method based on a non-hyperparameter total variation equilibrium constraint, and comprises the following steps: first, a total variation image denoising model based on columnar processing is established; then, a total variation image denoising cost function is constructed; then, according to a covariance fitting criterion, an optimal equilibrium weighting matrix of the total variation sparse constraint is derived, and a non-hyperparameter equilibrium total variation image denoising cost function is obtained; finally, optimal iterative solution is realized through a convex optimization tool. The method solves the non-hyperparameter equilibrium total variation image denoising cost function, realizes uniform denoising, does not lose the denoising effect, and solves the optimal selection problem of the regularization parameter in the existing total variation denoising method.

Description

Technical Field

[0001] This invention belongs to the field of image processing technology, specifically relating to an image denoising method based on total variational equalization without hyperparameter constraints. Background Technology

[0002] In the fields of image processing and computer vision, image denoising is a fundamental and important research problem, playing a vital role in applications such as image feature extraction, image recognition, and object detection.

[0003] The paper "Rudin LI, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D: nonlinear phenomena, 1992, 60(1-4): 259-268" proposes an image denoising method based on total variation norm. This method iteratively approximates the original image to the ideal denoised image, effectively preserving edge information. However, the reconstruction result exhibits a staircase effect. The paper "Patil P D, Kumbhar AD. Bilateral filter for image denoising 2015 International Conference on Green Computing and Internet of Things (ICGCIoT). IEEE, 2015: 299-302" proposes an image denoising method based on a bilateral filter. This method is effective with low-dose Gaussian noise, but its denoising performance rapidly decreases with increasing noise levels. The paper "Ghulyani M, Arigovindan M. Fast total variation based image restoration under mixed Poisson-Gaussian noise model 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018). IEEE, 2018: 1264-1267" proposes a fast image denoising algorithm based on mixed Poisson-Gaussian likelihood function and TV regularization (ADMM). It improves the convergence speed of the entire ADMM iteration by using variable segmentation. However, this method requires manually adjusted regularization parameters, which is not conducive to the implementation of engineering applications. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides an image denoising method based on total variational equalization without hyperparameter constraints. This method solves the problem of optimal selection of regularization parameters in existing total variational image denoising methods and greatly expands the application scope.

[0005] The technical solution adopted in this invention is: an image denoising method based on total variational equalization without hyperparameter constraints, the specific steps of which are as follows:

[0006] Step 1: Establish a noisy image model and perform columnar processing on the model;

[0007] The noisy image signal model in image processing is represented as follows:

[0008] Y = AX + E (1)

[0009] in, Represents a noisy image matrix. Represents the original image matrix. Represents a dictionary matrix. Represents the additive noise matrix; The vector space dimension is represented by M and N, which represent the size of the matrix rows and columns, respectively.

[0010] The model is columnarized, and three matrices F1, F2, and F3 are assumed to exist. According to matrix columnarization theory, we can obtain:

[0011]

[0012] Where vec(·) represents the matrix columnization operation, This represents the Kronecker product, (·). T This represents the matrix transpose operation.

[0013] Equation (1) can be transformed into a linear model:

[0014] y = Hx + e (3)

[0015] Where y = vec(Y), x = vec(X), e = vec(E), Denotes the identity matrix, and

[0016] Step 2: Construct the total variation image denoising cost function;

[0017] Adding both sparsity constraints and total variation constraints to the rows and columns of the linear model in equation (3) yields the total variation image denoising cost function, which is specifically expressed as:

[0018]

[0019] Where, x′=vec(X T), where λ represents the regularization parameter, and D represents the difference matrix:

[0020]

[0021] Step 3: Selection of the optimal equilibrium weighting matrix for total variation sparse constraints;

[0022] Based on the covariance fitting criterion, the optimal weighting matrix of the total variation sparse constraint is constructed to remove the regularization parameter and obtain the optimal solution.

[0023] First, let's rephrase equation (4) as follows:

[0024]

[0025] Next, let B = HD -1 B = [b1, b2, ..., b MN Then the weighted matrix W can be expressed as:

[0026] W = diag([w1, w2, ..., w MN (7)

[0027]

[0028] Here, diag(·) represents converting a vector into a diagonal matrix.

[0029] Weighted matrices W and D -1 There is a correlation; inverting the difference matrix D yields:

[0030]

[0031] Because of D -1 Due to the special structure, a difference matrix D′ is introduced to construct an equalization weighting matrix to balance the denoising effect. D′ is specifically represented as:

[0032]

[0033] The inverse matrix of D′ can be represented as:

[0034]

[0035] Let B′ = HD′ -1 , B′=[b′1,b′2,...,b′ M Then the equilibrium weighting matrix W′ can be expressed as:

[0036] W′=diag([w′1,w′2,...,w′ MN (13)

[0037]

[0038] Step 4: Construct a cost function for image denoising without hyperparameter equalization of total variation;

[0039] Based on the optimal equilibrium weighting matrix obtained in step three, the cost function for denoising a hyperparameter-free balanced total variation image under uniform noise conditions, obtained using the covariance fitting criterion, is:

[0040]

[0041] Step 5: Solve the model;

[0042] The model obtained in step four is a convex optimization problem, which can be solved quickly using the CVX toolbox in Matlab. The CVX solution result is as follows:

[0043]

[0044] in, The result is represented by CVX(·), and the solution process is represented by CVX Toolbox.

[0045] Finally, by solving the cost function of denoising a hyperparameter-free, fully variable image, uniform denoising is achieved while avoiding the problem of optimal selection of regularization parameters.

[0046] The beneficial effects of this invention are as follows: First, the method of this invention establishes a total variation image denoising model based on columnar processing. Then, it constructs a total variation image denoising cost function. Next, based on the covariance fitting criterion, it derives the optimal equilibrium weighting matrix for total variation sparse constraints, and obtains a hyperparameter-free balanced total variation image denoising cost function. Finally, it achieves the optimal iterative solution using a convex optimization tool. By solving the hyperparameter-free balanced total variation image denoising cost function, this invention achieves uniform denoising without sacrificing denoising effect, while solving the problem of optimal selection of regularization parameters in existing total variation denoising methods. Attached Figure Description

[0047] Figure 1 This is a flowchart of an image denoising method based on total variational equalization without hyperparameter constraints according to the present invention.

[0048] Figure 2 This is a schematic diagram of the original image used in an embodiment of the present invention.

[0049] Figure 3 This is a schematic diagram of a noisy image after noise addition in an embodiment of the present invention.

[0050] Figure 4 The image shows the result of a conventional total variation image denoising method in an embodiment of the present invention.

[0051] Figure 5The image shows the result of the hyperparameter-free total variation image denoising method in an embodiment of the present invention.

[0052] Figure 6 The image shows the result of the image denoising method without hyperparameter total variational equalization constraints in an embodiment of the present invention. Detailed Implementation

[0053] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0054] This invention uses simulation based on measured data to demonstrate the effectiveness of the proposed method. All steps and conclusions of this invention are verified on the Matlab2020b simulation platform.

[0055] like Figure 1 The flowchart of an image denoising method based on total variational equalization without hyperparameter constraints of the present invention is shown below. The specific steps are as follows:

[0056] Step 1: Establish a noisy image model and perform columnar processing on the model;

[0057] The noisy image signal model in image processing can be represented as:

[0058] Y = AX + E (1)

[0059] in, Represents a noisy image matrix. Represents the original image matrix. Represents a dictionary matrix. Represents the additive noise matrix; The vector space dimension is represented by M and N, which represent the size of the matrix rows and columns, respectively.

[0060] In this embodiment, the method of the present invention is used to denoise an image with added Gaussian noise.

[0061] Where M = 50, N = 50, A is the identity matrix, and Y is an image with additive Gaussian noise added, with a mean of 0 and a variance of 0.01.

[0062] In order to solve equation (1) using an optimization method, the model needs to be columnarized. Assuming there are three matrices F1, F2, and F3, according to matrix columnarization theory, we can obtain:

[0063]

[0064] Where vec(·) represents the matrix columnization operation, This represents the Kronecker product, (·). T This represents the matrix transpose operation;

[0065] Equation (1) can be transformed into a linear model:

[0066] y = Hx + e (3)

[0067] Where y = vec(Y), x = vec(X), e = vec(E), Denotes the identity matrix, and

[0068] A linear model is obtained by columnarization, and this model can be solved using optimization methods.

[0069] Step 2: Construct the total variation image denoising cost function;

[0070] Adding both sparsity constraints and total variation constraints to the rows and columns of the linear model in equation (3) yields the total variation image denoising cost function, which is specifically expressed as:

[0071]

[0072] Where, x′=vec(X T ), where λ represents the regularization parameter used to balance the constraint terms, and D represents the difference matrix:

[0073]

[0074] In equation (4), the regularization constraint on the row and column data of the image is achieved simultaneously through ||Dx||1+||Dx′||1. Choosing a suitable regularization parameter λ is necessary to solve this model, which is extremely difficult in practical engineering applications.

[0075] In this embodiment, the regularization parameter λ = 0.0224 was selected using the cross-validation method.

[0076] Step 3: Selection of the optimal equilibrium weighting matrix for total variation sparse constraints;

[0077] According to the covariance fitting criterion, the regularization parameter can be removed by constructing the optimal weighted matrix of the total variation sparse constraint to obtain the optimal solution.

[0078] First, let's rephrase equation (4) as follows:

[0079]

[0080] Next, let B = HD -1 B = [b1, b2, ..., b MN Then the weighted matrix W can be expressed as:

[0081] W = diag([w1, w2, ..., w MN (7)

[0082]

[0083] Here, diag(·) represents converting a vector into a diagonal matrix.

[0084] Weighted matrices W and D -1 There is a correlation; inverting the difference matrix D yields:

[0085]

[0086] Calculate the elements w in the weighted matrix W k At that time, w k The numerical value of D -1 The L2 norm of each column is related to the number of columns in (10). Observing the elements of each column in (10), it can be seen that as the number of columns k increases, w k It will also gradually increase, resulting in an imbalance in image denoising effects.

[0087] To address the above issues, this invention introduces a difference matrix D′ to construct an equalization weighting matrix to balance the denoising effect. Specifically, D′ is represented as:

[0088]

[0089] The inverse matrix of D′ can be represented as:

[0090]

[0091] Let B′ = HD′ -1 , B′=[b′1,b′2,...,b′ M Then the balanced weighting matrix W′ can be expressed as:

[0092] W′=diag([w′1,w′2,...,w′ MN (13)

[0093]

[0094] Step 4: Construct a cost function for image denoising without hyperparameter equalization of total variation;

[0095] Based on the optimal equilibrium weighting matrix obtained in step three, the cost function for denoising a hyperparameter-free, balanced total variation image can be obtained using the covariance fitting criterion under uniform noise conditions:

[0096]

[0097] Equation (15) avoids the selection of hyperparameters by introducing the optimal balanced weighting matrix, and also solves the problem of uneven image denoising results.

[0098] Step 5: Solve the model;

[0099] The model obtained in step four is a convex optimization problem, which can be solved quickly using the CVX toolbox in Matlab. The CVX solution result is as follows:

[0100]

[0101] in, The result is represented by CVX(·), and the solution process is represented by CVX Toolbox.

[0102] Finally, by solving the cost function of denoising a hyperparameter-free, fully variable image, uniform denoising is achieved while avoiding the problem of optimal selection of regularization parameters.

[0103] In this embodiment, to demonstrate the effectiveness of the method of the present invention, the proposed image denoising method based on total variational equilibrium without hyperparameter constraints is used to denoise an image with added Gaussian noise. The experiment was conducted on a 64-bit Matlab 2020b platform, and the solution method used MATLAB's CVX toolbox.

[0104] The original images used in the experiment are as follows Figure 2 As shown, its dimensions are 50×50. Gaussian noise with a mean of 0 and a variance of 0.01 is added to the image, resulting in a noisy image as shown. Figure 3 As shown. Figure 4 The solution results of existing total variation image denoising methods are presented, in which the parameter λ = 0.0224 is selected using the cross-validation method. Figure 5 The solution results are for a hyperparameter-free total variation image denoising method. Figure 6 The solution results are for the image denoising method without hyperparameter total variational equalization constraints proposed in this invention. Figure 4 , Figure 5 , Figure 6 The comparison shows that existing total variation image denoising methods can achieve good denoising results, but require complex selection of hyperparameters; while hyperparameter-free total variation image denoising methods can avoid hyperparameter selection, but there is uneven denoising results in the upper left corner of the image; the method of the present invention can not only avoid the selection of hyperparameters, but also solve the problem of uneven denoising results and reduce the loss of image details.

[0105] To quantitatively evaluate the denoising effect, the Peak Signal-to-Noise Ratio (PSNR) of each method compared to the original image was used. A higher PSNR indicates that the image is closer to the original image, and the better the denoising effect. The PSNR values ​​of each method are shown in Table 1.

[0106] Table 1

[0107] method Peak signal-to-noise ratio Noisy image 68.442 Existing total variation image denoising methods 73.5595 Hyperparameter-free total variation image denoising method 73.0663 Image denoising methods without hyperparameter total variational equalization constraints 73.8400

[0108] As can be seen from the table, the denoising capability of the method of the present invention is not only superior to the solution results of existing total variation image denoising methods, but also superior to the solution results of total variation image denoising methods without hyperparameters, thus solving the problem of uneven denoising results. Furthermore, the method of the present invention does not require the selection of hyperparameters during the calculation process, which greatly expands the application scope of the method.

[0109] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the invention should be included within the scope of the claims of the invention.

Claims

1. An image denoising method based on total variational equalization without hyperparameter constraints, the specific steps of which are as follows: Step 1: Establish a noisy image model and perform columnar processing on the model; The noisy image signal model in image processing is represented as follows: Y = AX + E (1) in, Represents a noisy image matrix. Represents the original image matrix. Represents a dictionary matrix. Represents the additive noise matrix; The vector space dimension is represented by M and N, which represent the size of the matrix's rows and columns, respectively. The model is columnarized, and three matrices F1, F2, and F3 are assumed to exist. According to matrix columnarization theory, we can obtain: Where vec(·) represents the matrix columnization operation, This represents the Kronecker product, (·). T This represents the matrix transpose operation; Equation (1) can be transformed into a linear model: y = Hx + e (3) Where y = vec(Y), x = vec(X), e = vec(E), Denotes the identity matrix, and Step 2: Construct the total variation image denoising cost function; Adding both sparsity constraints and total variation constraints to the rows and columns of the linear model in equation (3) yields the total variation image denoising cost function, which is specifically expressed as: Where, x′=vec(X T ), where λ represents the regularization parameter, and D represents the difference matrix: Step 3: Selection of the optimal equilibrium weighting matrix for total variation sparse constraints; Based on the covariance fitting criterion, the optimal weighting matrix of the total variation sparse constraint is constructed to remove the regularization parameter and obtain the optimal solution; First, let's rephrase equation (4) as follows: Next, let B = HD -1 B = [b1, b2, ..., b MN Then the weighted matrix W can be expressed as: W=diag([w1,w2,...,w MN ]) (7) Where, diag(·) represents converting a vector into a diagonal matrix; Weighted matrices W and D -1 There is a correlation; inverting the difference matrix D yields: Because of D -1 Due to the special structure, a difference matrix D′ is introduced to construct an equalization weighting matrix to balance the denoising effect. D′ is specifically represented as: The inverse matrix of D′ can be represented as: Let B′ = HD′ -1 , B′=[b′1,b′2,...,b′ M Then the equilibrium weighting matrix W′ can be expressed as: W′=diag([w′1,w′2,...,w′ MN ]) (13) Step 4: Construct a cost function for image denoising without hyperparameter equalization of total variation; Based on the optimal equilibrium weighting matrix obtained in step three, the cost function for denoising a hyperparameter-free balanced total variation image under uniform noise conditions, obtained using the covariance fitting criterion, is: Step 5: Solve the model; The model obtained in step four is a convex optimization problem, which can be solved quickly using the CVX toolbox in Matlab. The CVX solution result is as follows: in, The expression represents the solution result, and CVX(·) represents the solution process of the CVX toolbox. Finally, by solving the cost function of denoising a hyperparameter-free, fully variable image, uniform denoising is achieved while avoiding the problem of optimal selection of regularization parameters.

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